The 20-year-old Serbian winger moved to Anfield from Benfica for £20million last month and could make his debut live on Monday Night Football when Liverpool face Manchester City at the Etihad Stadium.
The Reds are set to sign complete a £16m deal for controversial striker Mario Balotelli, who will not be ready feature against his former club, but Markovic has backed himself to take Liverpool by storm.
“I was voted one of the best players in Serbia and Portugal and I hope I will be recognised as one of the best in the Premier League - I believe I will,” he said.
“Liverpool are one of the best clubs in the world and I felt last season they were the strongest team in the Premier League. It was bad luck that they didn't win it.
“I watched Liverpool all the time last season. They were very good and luck played its part as to why they didn't win the league. They should have been champions.
“The way the Liverpool fans are appealed to me. I have been lucky enough to have been at Partizan and Benfica, two clubs with passionate supporters.”
Markovic began his career at Partizan Belgrade before signing for Benfica last summer, where he scored six goals in 39 appearances in all competitions.